Review sheet for the secon d exam

• The idea of an infinite sequence What is a sequence? Why does it basically boil down to a function from the integers to the reals? How can sequences be combined?
• The idea of an infinite series What's the difference between a sequence and a series? Why does the notion of the "sum" of a series need to be defined? (That is, why isn't it just a problem for calculation instead of fundamental definition?)
• The limit definition of convergence of a sequence In particular, if you can calculate the n required to get within a certain epsilon of the supposed limit L, and if you can confirm that all larger n work too, then you have proven convergence. But more generally, you need to understand the logical meaning of the definition of convergence, in particular why is says for all m greater than or equal to n.
• Bounded increasing sequences converge, and why Because we say they do. In particular, because we assume that every nonempty bounded subset of the reals has a least upper bound, called the "lub" or "sup", depending who you talk to. How do we know that? We assume that. This is the last axiom of the real number system required to understand how numbers work. We apologize for withholding it for so long, but I guess your gradeschool teachers didn't think you needed to know.
• Rules of limits (what limits pass across, and when) Limits pass across basically everything, including, importantly, continuous functions. The even pass across quotients, but only if the result of passing them across makes sense. (e.g., no arithmetic with infinities, no zeroes in denominators.) For more complicated quotients, L'Hopital is required.
• The definition of series convergence in terms of partial sums of sequences. If you understand why jason is so preoccupied with partial sums, then you'll understand the definitino of the infinite sum of an infinite series. It is odd, but until we define the infinite sum in this way, the infinite addition problem isn't actually sensible, because it's a request for arithmetic to do what it has never done before.
• L'Hopital's rule for solving difficult limit problems Two cautions: First, don't get confused between L'Hopital and the quotient rule. Second, and more importantly, L'Hopital only works when you have an indeterminate form in the first place. So any attempt to apply L'Hopital "just in case" to a problem which is not already "hard" will probably just give you the wrong answer. Legend tells of trick questions on exams based on this principle.
• Dealing with difficult indeterminate forms via L'Hopital's rule. Exponents get converted to products via the logarithm function. Products get converted to quotients via inversion. Sums get converted either to products by factoring or to quotients by multiplying and dividing by something clever.
• Improper integrals, and how to solve them using limits. Arithmetic with infinity is bound to get you into trouble. But you can nevertheless solve an integral all the way to infinity by changing the bound, and taking a limit. Points off for plugging in infinity. Also, many integrals which appear innocent enough are in fact improper because they can have asymptotes within their domain. Please be on the lookout for these, because stories of old tell of devious professors giving exam questions based on this principle.
• Improper integrals in applications. Applications to energy as the integral of force, or for that matter, anything at the integral of anything, physically. We talked about escape velocity and electrons colliding with protons, one of which made perfect sense, and one of which turned out to be nonsense based on bad physical assumptions (point particles aren't quite right).
• The integral test for series convergence. The integral is not the sum. The integral is not the sum. The integral is not the sum. But they're so close to each other (differing by a finite amount) that if you can for some reason actually calculate the integral (which will be improper), then you can use the answer to detect whether the sum converges or diverges. This little fact means that integrals may crop up where you least expect them.
• The comparison test for series convergence. And using it effectively, which is harder than it sounds.
• The comparison test for series divergence. ditto.
• The ratio test for series divergence. This requires that you take a limit of a sequence. Normally, limits of sequences are easy to take, but keep in mind that if your professor is really devious, such a step might require power tools, like L'Hopital's rule.
• Alternating series, and conditional and absolute convergence. A'la Dr. Zucker's lecture.
• Geometric series, and calculating their sums. 1/(1-r), for r between -1 and 1, obviously. But also be on the lookout. It's important to notice if a given series at hand is geometric.
• The harmonic series, and its sum. It's sum is infinity, as you can show by the integral test or by the grouping argument in class. This is a very important example, because it teaches you that even if the terms of a sum go to zero, the sum may not converge.